Time-independent Schrodinger equation \(\rightarrow\) ODE: \[ \begin{equation} \left[-\frac{\hbar^2}{2m}\frac{\mathrm{d}^2}{\mathrm{d}x^2} + V(x)\right] \Psi(x) = E \Psi(x) \rightarrow \left[\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \frac{2m}{\hbar^2} (E - V_i) \right] \Psi_i(x) = 0. \end{equation} \]
potential step — \(V(x)<E\) everywhere
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potential barrier — \(V_b>E\)
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potential barrier — \(V_b>E\)
\[ \begin{align} &k_1=\sqrt{\frac{2m}{\hbar}^2E}\\ &\Psi_1(x) = A_1 e^{i k_1 x} + B_1 e^{-i k_1 x} \end{align} \]
\[ \begin{align} &\kappa_2=\sqrt{\frac{2m}{\hbar}^2(V_b-E)},\\ &\Psi_2(x) = A_2 e^{\kappa_2 x} + B_2 e^{-\kappa_2 x}, \end{align} \]
\[ \begin{align} &k_3=\sqrt{\frac{2m}{\hbar}^2E}\\ &\Psi_3(x) = A_3 e^{i k_3 x} + B_3 e^{-i k_3 x} \end{align} \]
\(\Psi_1(0) = \Psi_2(0)\) \(\Psi_1'(0) = \Psi_2'(0)\)
\(\Psi_2(L) = \Psi_3(L)\) \(\Psi_2'(L) = \Psi_3'(L)\)
ODE solutions: \[ \begin{align} &\Psi_1(x) = e^{i k_1 x} + \color{gray}{B_1} e^{-i k_1 x},\\ &\Psi_2(x) = \color{gray}{A_2} e^{\kappa_2 x} + \color{gray}{B_2} e^{-\kappa_2 x},\\ &\Psi_3(x) = \color{gray}{A_3} e^{i k_3 x} \end{align} \]
…with boundary conditions: \[ \begin{align} \Psi_1(0) = \Psi_2(0),\\ \Psi_1'(0) = \Psi_2'(0),\\ \Psi_2(L) = \Psi_3(L),\\ \Psi_2'(L) = \Psi_3'(L) \end{align} \]
…algebraic problem! \[\begin{equation} \begin{pmatrix} 1 & -1 & -1 & 0 \\ -ik_1 & -\kappa_2 & \kappa_2 & 0\\ 0 & e^{\kappa_2 L} & e^{-\kappa_2 L} & -e^{ik_3 L} \\ 0 & \kappa_2 e^{\kappa_2 L} & -\kappa_2 e^{-\kappa_2 L} & -ik_3 e^{ik_3 L} \end{pmatrix} \begin{pmatrix} B_1\\A_2\\B_2\\A_3 \end{pmatrix} = \begin{pmatrix} -1 \\ -ik_1 \\ 0 \\ 0 \end{pmatrix} \end{equation}\]
solve with Wolfram Mathematica?
M = {{
1,-1,-1,0},
{-\[ImaginaryI],-\[Kappa]2/k1,\[Kappa]2/k1,0},
{0,Exp[\[Kappa]2 L],Exp[-\[Kappa]2 L],-Exp[\[ImaginaryI] k3 L]},
{0,Exp[\[Kappa]2 L],- Exp[-\[Kappa]2 L],-\[ImaginaryI] k3/\[Kappa]2 Exp[\[ImaginaryI] k3 L]}
};
b = {-1, -\[ImaginaryI],0,0};
ln = LinearSolve[M,b];
B1 = ln[[1]]//FullSimplify
A2 = ln[[2]]//FullSimplify
B2 = ln[[3]]//FullSimplify
A3 = ln[[4]]//FullSimplify
wavefunctions \(\Psi(x)\) incident on a potential barrier
- \(V_b = 4\),
- \(E = 2\),
- \(L \kappa_2 = 0.3\).
wavefunctions \(\Psi(x, t) = \exp(-iE t/\hbar)\Psi(x)\) incident on a potential barrier
- \(V_b = 4\),
- \(E = 2\),
- \(L \kappa_2 = 0.3\).
wavefunctions \(\Psi(x, t) = \exp(-iE t/\hbar)\Psi(x)\) incident on a potential barrier
- \(V_b = 4\),
- \(E = 2\),
- \(L \kappa_2 = 1\).
wavefunctions \(\Psi(x, t) = \exp(-iE t/\hbar)\Psi(x)\) incident on a potential barrier
- \(V_b = 4\),
- \(E = 2\),
- \(L \kappa_2 = 2\).
wavefunctions \(\Psi(x, t) = \exp(-iE t/\hbar)\Psi(x)\) incident on a potential barrier
- \(V_b = 4\),
- \(E = 2\),
- \(L \kappa_2 = 8\).
transmission \(\mathcal{T}\) can be calculated as (see discussion on current densities in the lecture notes)
\[ \begin{equation} \color{green}{\mathcal{T}} = \left|\frac{2 e^{-ik_3 L}k_1 \kappa_2}{(k_1+k_3)\kappa_2 \cosh(L \kappa_2) + i(- k_1 k_3+\kappa_2^2)\sinh(L \kappa_2)}\right|^2, \end{equation} \]
for thick and/or tall barriers (\(L\kappa_2 \gg 1\)) (homework problem)
\[\begin{equation} \mathcal{T} \approx 4 e^{-2\kappa_2 L}\left[1+\left(\frac{\kappa_2^2-k_1^2}{2k_1\kappa_2}\right)^2\right]^{-1}. \end{equation} \]
for thin barriers and/or energies right below the barrier (\(L\kappa_2 \ll 1\)) (homework problem) \[ \begin{equation} \mathcal{T} = |A_3|^2 \approx 1- (L\kappa_2)^2\left[1+\left(\frac{\kappa_2^2-k_1^2}{2k_1\kappa_2}\right)^2\right]. \end{equation} \]
